proceedings - dublin city university · 2020. 9. 7. · proceedings of 24 th and 25 th september,...

Proceedings of

24 th and 25 th September, 2009

St. Patrick’s College, Drumcondra, Dublin 9

Editors: Dolores Corcoran, Thérèse Dooley, Sean Close and Ronan Ward

St. Patrick’s College, Drumcondra, Dublin 9.

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TABLE OF CONTENTS

Acknowledgements 1

KEYNOTE ADDRESSES

Equity and high achievement: The case of Railside school Jo Boaler, University of Sussex

2

Teachers assessing young children’s mathematical development: How confident are they?

Janette Bobis, University of Sydney, Australia

20

Mathematics teaching as a shared enterprise: Learning by ‘doing’

Dolores Corcoran, St Patrick’s College, Drumcondra

34

Maths for Al Jan van Maanen, Freudenthal Institute, Utrecht University (NL)

53

PRIMARY/EARLY YEARS MATHEMATICS EDUCATION

Realistic Mathematics Education in an Irish primary classroom

Patricia Cassidy, Gardiner Street Primary School, Dublin

67

Characteristics and curriculum validity of test items for the National Assessment of Mathematics Achievement in 6 th class

Seán Close, David Millar and Gerry Sheil, Educational Research Centre, St Patrick’s College

77

Mathematical knowledge for teaching 3D shapes

Seán Delaney, Cólaiste Mhuire, Marino Institute of Education

92

Insight in primary mathematics: Teacher ‘moves’ that facilitate the ‘stumbling of ideas across each other’

Thérèse Dooley, St Patrick’s College, Drumcondra & University of Cambridge, UK

104

Do Mathematics textbooks or workbooks enhance the teaching of mathematics in early childhood?: Views of teachers of four and fiveyearold children in primary schools in Ireland

Elizabeth Dunphy, St Patrick’s College, Drumcondra

114

The use of Mathematics textbooks to promote understanding in the lower primary years

Lorraine Harbison, University of Newcastle upon Tyne

123

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Working towards addressing the Mathematics subject matter knowledge needs of prospective teachers

Mairead Hourigan, Mary Immaculate College and John O’Donoghue, University of Limerick

133

Teaching fractions in primary school: How is a teachers’ knowledge communicated to pupils?

Bodil Kleve, Oslo University College, Norway

144

Exploring the use of Numicon in a mainstream primary classroom in the Republic of Ireland

Marie Lane, University College Dublin

154

A picture is worth a thousand words: Insights into graphicacy skills of primary prospective preservice teachers

Aisling M. Leavy, Mary Immaculate College

164

Four years later

Yvonne Mullan, National Educational Psychological Service

177

Endeavouring to teach mathematical problem solving from a constructivist perspective

John O’Shea, Mary Immaculate College

185

Learning support for Mathematics: Lessons from 100 lessons

Joseph Travers, St Patrick’s College, Drumcondra

195

“I thought this was a trick question”Realistic mathematical modelling: A class study

Ronan Ward, St Patrick’s College, Drumcondra

208

POSTPRIMARY MATHEMATICS EDUCATION

“Applicable Mathematics” in seniorcycle mathematics education: Selected results of an Irish research project

Brian Carroll, NCEMSTL and John O’Donoghue, University of Limerick

218

Who am I and how did get here?: Exploring the mathematical identity of student teachers

Patricia Eaton, Stranmillis University College, Belfast and Maurice OReilly, St Patrick’s College, Drumcondra

228

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The importance of preservice teachers’ conceptions of Mathematics and approaches to learning for the future of mathematics education in Ireland

Miriam Liston and John O’Donoghue, NCEMSTL

238

Why it’s different with Mathematics: Prospective teachers’ reflections on what makes teaching postprimary Mathematics unique

Maria Meehan, University College Dublin and Catherine Paolucci, NUI Galway

250

Reform via textbooks: Lessons from a crossnational collaboration

Pamel Moffett, Stranmillis University College, Belfast

262

Assessing the level of suitably qualified teachers teaching Mathematics at post primary education in Ireland

Máire Ní Ríordáin, NCEMSTL and Ailish Hannigan, University of Limerick

273

Assessing the effect of Mathematics textbook content structure on student comprehension and motivation

Lisa O’Keeffe, NCEMSTL and John O’Donoghue, University of Limerick

283

Solving problems in Mathematics education: Challenges for Project Maths

Elizabeth Oldham, Trinity College, University of Dublin and Sean Close, Educational Research Centre, St Patrick’s College

295

The ladder of knowledge: Knowledge for effective teaching

Niamh O’Meara, University of Limerick and John O’Donoghue, NCEMSTL

309

Continuous professional development in mathematics education: Considerations for the Irish context

Mark Prendergast, University of Limerick and John O’Donoghue, NCEMSTL

321

THIRDLEVEL MATHEMATICS EDUCATION

Constructing and validating an instrument to measure students’ attitudes and beliefs about learning Mathematics

Sinead Breen, St Patrick’s College, Drumcondra, Joan Cleary, Institute of Technology, Tralee and Ann O’Shea, NUI Maynooth

332

Determining the validity of mathematical statements in a thirdlevel Analysis course

Nuala Curley and Maria Meehan, University College Dublin

343

The changing profile of thirdlevel service Mathematics students (19972008)

Fiona Faulkner, University of Limerick, Ailish Hannigan, University of Limerick and Olivia Gill, NCEMSTL

355

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Diagnostic testing in DCU: A fiveyear review

Eabhnat Ní Fhloinn, Dublin City University

367

Extending the mathematical capacity of Gaeilgeoirí: Assessing the effectiveness of bilingual Mathematics instruction in first year undergraduate education

Máire Ní Ríordáin, NCEMSTL and Aisling McCluskey, NUI Galway

379

The power of the short story and the big picture in mathematics education in schools, universities and for the general public

Fiacre Ó Cairbre, NUI Maynooth

391

The role of proof validation in students’ mathematical learning

Kirsten Pfeiffer, NUI Galway

403

What makes Mathematics attractive at university?

Rachel Quinlan, NUI Galway

414

Proofs, wranglers and virtual conversations

Tim Rowland, University of Cambridge, UK

425

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ACKNOWLEDGEMENTS

We wish to thank the keynote speakers, presenters of research reports, and all participants in the third MEI conference on Research in Mathematics Education in Ireland. Our main aim – that of searching for ways to achieve more equitable outcomes for all students of mathematics – is reflected in our conference theme, ‘Mathematics for All: Extending Mathematical Capacity’. In order to achieve this, we seek to bring together those who have an interest in mathematics education, to consider future directions for mathematics education research in Ireland in the light of recent developments and international trends, and to consider ways of improving linkages with mathematics education communities within Ireland and in other countries. We hope that we succeed in this and that the conference provides participants with an interesting and challenging programme of presentations, and with opportunities for discussion and making contacts and renewing friendships. We are happy that Irish mathematics education is flourishing and we welcome links with the wider scientific research community.

We also express our sincere gratitude to all those who supported the conference including: Dr. Pauric Travers, President of St. Patrick’s College; Professor Kathleen Lynch, UCD, and our sponsors St. Patrick’s College Research Committee, the Department of Education and Science, The Teachers’ Council, Irish National Teachers’ Organization and the Centre for the Advancement of Science Teaching and Learning (DCU).

Thérèse Dooley (Chairperson, MEI3 Organising Committee)

ORGANISING COMMITTEE Sinead Breen

Seán Close Dolores Corcoran

Thérèse Dooley Elizabeth Dunphy

Maurice O’Reilly Joe Travers

Ronan Ward

Proceedings of Third National Conference on Research in Mathematics Education

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EQUITY AND HIGH ACHIEVEMENT: THE CASE OF RAILSIDE SCHOOL Jo Boaler

University of Sussex

The low and inequitable mathematics performance of students in urban schools has been identified as a critical issue for our times. This paper will report the results of a fouryear study of approximately 700 students as they progressed through three different Californian high schools. One of the findings was the important success of “Railside” high school where the teachers taught mixed ability classes and used a particular pedagogical approach called ‘complex instruction’. The students at Railside learned more mathematics, enjoyed mathematics more and progressed to higher levels. In this paper I will detail the teachers’ methods and the impact they had upon the students.

INTRODUCTION

In order to consider the impact of different teaching approaches upon students’ understanding of mathematics, a team of graduate students and I conducted a longitudinal fouryear study of three high schools. The study involved monitoring approximately 700 students as they progressed through four years of three schools that were chosen because they offered different mathematics teaching approaches. The school that is the focus of this paper was given the pseudonym of ‘Railside’, as it was located next to the railway tracks in an urban setting in California. The students were from diverse ethnic and cultural groups and were largely from lowincome homes. The other two schools that we called ‘Greendale’ and ‘Hilltop’ were in more suburban settings with less ethnic diversity. Greendale school had little ethnic diversity and almost all students were white; at Hilltop school most students were white or Hispanic. Table 1 gives details of the three schools and their populations of students. In order to monitor and analyze the teaching approaches in the three schools we observed over 600 hundred hours of lessons, many of which were videotaped. These lessons were analyzed in different ways as set out below. In addition, we interviewed students in every year of the study to consider their reported experiences and interpretations of mathematics class. Students were typically interviewed in same sex pairs and we interviewed approximately sixty students each year, sampling high and low achievers from each approach in every school, taking care to interview students from different cultural and ethnic groups. In addition to interviews, we also administered questionnaires to all the students in the focus cohorts in years 1, 2 and 3 of the study (when most students were required to take mathematics). The questionnaires combined closed, Likert response questions, with more open questions that we analyzed and coded. The questionnaires asked students about their experiences in class, their enjoyment of mathematics, their perceptions about the nature of mathematics, learning, and students.

Mathematics for All — Extending Mathematical Capacity

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Table 1: Schools & Students School: Railside Hilltop Greendale

Approx school

population

1500 1900 1200

Study

Demographics

38% Latino/a,

23% African American,

20% White,

16% Asian or Pacific

Islanders.

3% other groups.

57% White,

38% Latino/a,

5% other ethnicities.

90% White,

5% Latino/a,

5% other ethnicities.

ELL students 25% 24% 0%

Free/reduced lunch 31% 23% 9%

Parent education,

% college

graduates

23% 33% 37%

The observations, interviews and questionnaires combined to give us information on the teaching and learning practices in the different approaches and students’ responses to them. Teachers from each approach were also interviewed at various points in the study although the teachers’ perspectives on their teaching were not a major part of our analyses. In addition to monitoring the students’ experiences of the mathematics teaching and learning, we also assessed their understanding in a range of different ways, administering tests to all students in the different school approaches as well as longer applied assessments that were given to focus groups and videotaped (Fiori & Boaler, 2004).

DATA ANALYSIS

Data from our classroom observations were analyzed in three different ways. First, we drew upon our observations from class visits and videotapes to produce ‘thick descriptions’ (Geertz, 2000) of the teaching and learning in the different classes. We collectively watched over 600 hours of lessons and different observers of the classes and the videotapes discussed and highlighted the most salient features of each approach and the differences between them. Second, we conducted a quantitative analysis of time spent in classes. This involved spending a year observing videotapes and deciding upon the different, mutually exclusive ways in which students spent time in class. These included such categories as ‘teacher talking’, ‘teacher questioning with whole class’, ‘students


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working alone’, and ‘students working in groups’. When agreement was reached on the categories that should be used, three researchers coded lessons until over 85% agreement was reached on the coding. We then completed the coding of over 55 hours of lessons, coding every 30second period of time. This yielded 6,800 coded segments. We also recorded the amount of time that was spent on each mathematics problem in class. This coding exercise was only performed on year one classes as it was extremely time intensive and we lacked the resources to perform the same analysis every year but qualitative observations of lessons suggested that similar differences between the different approaches pertained to each year of the study. In addition to these qualitative and quantitative analyses of lessons, we performed a detailed analysis of the questions teachers asked students. This level of analysis fell between the qualitative and quantitative methods we had used and was designed in response to our awareness that the teachers’ questions were an important indicator of the mathematics on which students and teachers worked (see Boaler & Brodie, 2004). Our coding of teacher questions was more detailed and interpretive than our coding of instructional time but it was sufficiently quantitative to enable comparisons across classes.

In order to analyze the detailed student interviews they were first read by teams of researchers and then coded (Glaser & Strauss, 1967; Miles & Huberman, 1994). Codes were first identified by different researchers using a process of open coding; the list of agreed upon codes was then used to recode all interviews. Questionnaires were analyzed quantitatively with both individual questions and scales of questions being subject to exploratory and confirmatory statistics. The assessments in our study ranged from tests that were scored, blind, and statistically analyzed, to problem solving sessions that were videotaped and assessed. For the videotaped assessments individual student work was graded and rubrics were developed and used to assess the interactions of groups as they worked (Fiori & Boaler, 2004). In addition to the individual analysis of each data source (lesson observations, interviews, videos, questionnaires, assessments) the findings from these multiple sources were then analyzed and understood in relation to one another, thus illuminating trends and themes across sources and affording the opportunity to triangulate the data. As data were analyzed by a team of researchers themes were discussed and agreed upon by groups of people, which served to increase confidence in our analyses and findings (Eisenhart, 2002). In addition, constant comparison across cases (Glaser & Strauss, 1967, 1991) was used to illuminate critical defining features and practices of each school. This allowed us to capture subtle aspects of each learning environment that may have otherwise been overlooked. The analyses were shared with the teachers as a form of member check (Glesne & Peshkin, 1992), further enhancing the validity of the findings. The most important teaching and learning interactions identified in this paper emerged from the observations of classes, the student interviews, and the questionnaires and they will be the main data sources reported in this paper.


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TIME SPENT AND ACHIEVEMENT

The majority of students in our study experienced one of two teaching approaches. The two suburban schools Greendale and Hilltop offered a choice between a ‘traditional’ sequence of courses taught traditionally and a ‘reform’ sequence taught using more open problems and groupwork, but few students chose the reform courses and we had insufficient numbers to include in our analyses. Most of the students at Greendale and Hilltop therefore experienced a traditional approach, as named by the schools. The teachers lectured and the students practiced methods, working their way through short questions. Our coding of videotapes allowed us to categorise the ways in which students spent time in their classes. This showed that approximately 21% of the time in ‘traditional’ algebra classes at Greendale and Hilltop was spent with teachers lecturing, usually demonstrating methods. Approximately 15% of the time teachers questioned students in a whole class format. Approximately 48% of the time students were practising methods in their books, working individually; approximately 11% of the time they worked in groups, and students presented work for approximately 0.2% of the time. The average time spent on each mathematics question was 2.5 minutes. The second major approach in our study was the one offered at Railside school in which the teachers posed longer, conceptual problems; students worked in groups and they often presented their work while teachers questioned presenters and other students. Our coding of videos showed that teachers lectured to classes for approximately 4% of the time. Approximately 9% of the time teachers questioned students in a whole class format. Approximately 72% of the time the students worked in groups while teachers circulated the room teaching methods and asking the students questions of their work, and students presented work for approximately 9% of the time. The average time spent on each mathematics problem at Railside was 5.7 minutes. An additional important difference between the schools was that Greendale and Hilltop employed ability grouping and students were placed into one of three different levels of classes at the beginning of high school. At Railside all students were placed into heterogeneous classes.

The students at Railside started high school at significantly lower mathematics levels than the students in the more suburban schools (t = 9.141, p


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(predominantly east Asian), Filipino, and White students were each outperforming Hispanic and Black students. At the end of Year 1, only one year after the students started at Railside, there were no longer significant differences between the achievement of White and Hispanic students, nor Filipino students and Hispanic and Black students. In subsequent years the only consistent difference that remained was the high performance of (East) Asian students who continued to significantly outperform Black and Hispanic students, but differences between White, Black and Hispanic students disappeared. Achievement differences between students of different ethnicities at Hilltop, where approximately half of the students were White and half Hispanic, remained with the White students outperforming Hispanic students, reflecting inequities that are fairly typical for urban schools in the US (Haberman, 1991; Kozol, 1992). At no time in the study were there any achievement differences by gender in any of the schools and girls and boys were represented equally in the different classes.

ANALYZING THE SOURCES OF SUCCESS

The Department, Curriculum and Timetable

Railside school has an unusual mathematics department. Twelve of the thirteen teachers work collaboratively, spending vast amounts of time designing curriculum, discussing teaching decisions and actions, and generally improving their practice through the sharing of ideas. A study conducted by Horn on the ways in which the department collaborate, found that the teachers spent around 650 minutes a week planning, individually and collectively (their paid work week provides 450 minutes of preparation time) (Horn, 2002). Unusually for the United States, the mathematics department strongly influences the recruitment and hiring of teachers, enabling the department to maintain a core of teachers with shared philosophies and goals. The teachers share a strong commitment to the advancement of equity. The department has spent many years working out a coherent curriculum and teaching approach, and they strive to ensure that good teaching practices are shared. One way in which this is achieved is through a practice that the department calls “following.” The cochairs structure teaching schedules so that a new teacher can stay a day or two behind a more experienced teacher, allowing the new teacher to observe lessons and activities during their daily preparation period before they try to adapt it for their classrooms (Horn, 2002; 2005). In addition, teachers share teaching practices and moves in weekly meetings in order that students experience a consistent approach.

The mathematics department has focused in particular upon the introductory algebra curriculum that all students take when they start the school. The algebra course is designed around key concepts with questions from various published curriculum such as CPM, IMP and a textbook of activities that use algebra Lab Gear (Picciotto & Wah, 1994). A theme of the algebra and subsequent courses is multiple representations, and students are frequently asked to represent their ideas in different ways, using words, graphs, tables and symbols. In addition, connections between algebra and geometry are


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emphasized even though the two areas are taught in separate courses. Railside follows a practice of ‘block scheduling’ and lessons are 90 minutes long, with courses taking place over half a school year, rather than a full academic year. In addition, the introductory algebra curriculum that is generally taught in one course in US high schools, is taught in the equivalent of two courses at Railside. The teachers have spread the introductory content over a longer period of time partly to ensure that the foundational mathematical ideas are taught carefully with depth and partly to ensure that particular norms – both social and sociomathematical (Yackel & Cobb, 1996) – are carefully established. The fact that mathematics courses are only half a year long at Railside may appear unimportant but in fact this organizational decision has a profound impact upon the students’ opportunities to take higherlevel mathematics courses. In most North American high schools mathematics classes are one year long and they begin with algebra. This means that students cannot take calculus unless they are advanced, as the typical sequence of courses is algebra, geometry, advanced algebra then precalculus. If a student fails a course at any time they are knocked out of that sequence and have to retake the course, further limiting the level of content they will reach. At Railside the students could take two mathematics classes each year. This meant that students could fail classes, start at lower levels, and/or choose not to take mathematics in a particular term and still reach calculus. This relatively simple scheduling decision is part of the reason that significantly more students at Railside took advanced levels classes at school than students in the other two schools.

Another important difference between the classes in the three schools we studied was the heterogeneous nature of Railside classes. Whereas incoming students in Greendale and Hilltop could enter geometry or could be placed in a remedial class, such as ‘math A’ or ‘business math’, all students at Railside entered algebra classes. The department is deeply committed to the practice of mixed ability teaching and to giving all students equal opportunities for advancement.

Recent years have pointed to the importance of school and district contexts in the support of teaching reforms. Such support is undoubtedly important but Railside is not a case of a district or school that initiated or mandated reforms. The reforms put in place by the mathematics department were in line with other school reforms but they were driven by the passion and commitment of the mathematics teachers in the department. The school, in many ways, provided a demanding context for the reforms, not least because they had been managed by five different principals in the six years we were there, and they had been labelled an 'underperforming' school by the state because of low state test scores. Neither of these factors is atypical in poorly resourced school districts in the US. The department fought to maintain their practices at various times and worked hard to garner the support of the district and school, and while the teachers felt well supported at the end of our study Railside does not represent a case of a reforming district encouraging a department to engage in new practices. Rather, Railside is a case of an unusual,


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committed and hard working department that continues to grow in strength through its teacher collaborations and work.

Groupwork and Complex Instruction

Some mathematics departments in the US employ group work with limited success, particularly because groups do not always function well, with some students doing more of the work than others, and some students being excluded or choosing to opt out. At Railside the teachers employed additional strategies to make group work successful. They adopted an approach called ‘complex instruction’ designed, by Liz Cohen and Rachel Lotan in the US (Cohen, 1994; Cohen & Lotan, 1997) for use in all subject areas. The system is designed to counter social and academic status differences in classrooms, starting from the premise that status differences do not emerge because of particular students but because of group interactions. The approach includes a number of recommended practices that the school employs, as I shall now describe:

Roles

When students are placed into groups they are also given a particular role to play, such as ‘facilitator’, ‘team captain’, ’recorder/reporter’ or ‘resource manager’ (Cohen & Lotan, 1997). The premise behind this approach is that all students have important work to do in groups, without which the group cannot function. At Railside the teachers emphasize the different roles at frequent intervals, stopping, for example, at the start of class to remind ‘facilitators’ to help people check answers or show their work or to ask the group: ‘what did you get for number 1?’. Students change roles at the end of each unit of work. The teachers reinforce the status of the different roles and the important part they play in the mathematical work that is being undertaken. Although I will not write more about the roles in this paper, they contribute to the complex interconnected system that operates in each classroom; a system in which everyone has something important to do and all students learn to rely upon each other. Readers may consult the literature on complex instruction (or visit www.complexinstruction.org) for more information on the roles.

Multidimensionality

In many mathematics classrooms there is one practice that is valued above all others – that of executing procedures (correctly and quickly). The narrowness by which success is judged means that some students rise to the top of classes, gaining good grades and teacher praise, whilst others sink to the bottom with most students knowing where they are in the hierarchy created. Such classrooms are unidimensional – the dimensions along which success is presented are singular. A central tenet of the complex instruction approach is what the authors refer to as ‘multiple ability treatment’. This ‘treatment’ is based upon the idea that expectations of success and failure can be modified by the provision of a more open set of task requirements that value many different ‘abilities’. Teachers should explain to students that ‘no one student will be “good on all these


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abilities” and that each student will be “good on at least one”’ (Cohen & Lotan, 1977, p. 78).

At Railside the teachers create multidimensional classes by valuing many dimensions of mathematical work. This is achieved – in part – by having more open problems that students can solve in different ways. The teachers value different methods and solution paths and this enables more students to contribute ideas and feel valued. But multiple solution paths are not the only contributions that are valued by teachers. When we interviewed the students and asked them “what does it take to be successful in mathematics class?” they offered many different practices such as: asking good questions, rephrasing problems, explaining well, being logical, justifying work, considering answers, and using manipulatives. When we asked students in ‘traditional’ classes what they needed to do in order to be successful they talked in much more narrow ways, with 97% of the students naming the same practice of “paying careful attention”.

The multidimensional nature of the classes at Railside was an extremely important part of the increased success of students. Put simply, when there are many ways to be successful, many more students are successful. Students are aware of the different practices that are valued and they feel successful because they are able to excel at some of them. Teachers at other schools may not encourage practices outside of procedure execution because they are not needed in state tests, but the fact that teachers at Railside valued a range of practices and more students could be successful in class made students feel more confident and positive about mathematics. This probably enhanced their success on tests even when tests assessed a more narrow range of mathematical work.

The following comments given by students in interviews give a clear indication of the multidimensionality of classes

Janet: Back in middle school the only thing you worked on was your math skills. But here you work socially and you also try to learn to help people and get help. Like you improve on your social skills, math skills and logic skills. (Year 1)

And:

Jasmine: With math you have to interact with everybody and talk to them and answer their questions. You can’t be just like “oh here’s the book, look at the numbers and figure it out”.

Interviewer: Why is that different for math? Jasmine: It’s not just one way to do it (…) It’s more interpretive. It’s not just one

answer. There’s more than one way to get it. And then it’s like: “why does it work”? (Year 1)

These students recognize that helping, interpreting and justifying are critically valued practices. The following student describes another valued dimension:


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Jorge: A math person is a person who knows like, how to do the work and then explain it. Like explaining everything to everyone so they could get it. Or they could explain it the hard way, the easy way or just, like average – so we could all get it. That’s like a math person I think. (Year 1)

Jorge shows that he appreciates something sophisticated – that explanations can vary and that a “math person” can explain in different ways. Given the frequent explanations these students give and hear, it may be unsurprising – but nevertheless important – that he appreciates the distinguishing qualities of different explanations.

One of the practices that I have come to regard as being particularly important in the promotion of equity, is justification. At Railside, students are required to justify their answers at almost all times. There are many good reasons for this – justification is an intrinsically mathematical practice (RAND, 2002; Martino & Maher, 1999), but this practice also serves an interesting and particular role in the promotion of equity. Many teachers struggle to deal with the wide range of students who attend classes, particularly in introductory classes such as high school algebra, which include students who are motivated with a wealth of prior knowledge as well as those who are less motivated and /or lack basic mathematical knowledge. Teachers want to help all of the students but the gap between the knowledge of lower attaining and higher attaining students can be very difficult to address. At Railside, school classes have as wide a gap as any I have seen but the teachers embrace the diversity they encounter and one practice that helps them support the learning of all students is justification. The following two students give some indication of the role of justification in helping different students:

Int: What happens when someone says an answer? Ana: We’ll ask how they got it

Latisha: Yeah because we do that a lot in class. (…) Some of the students – it’ll be the students that don’t do their work, that’d be the ones, they’ll be the ones to ask step by step. But a lot of people would probably ask how to approach it. And then if they did something else they would show how they did it. And then you just have a little session! (Year 3)

It is noteworthy that these two students do not talk about students who are slow, dumb or stupid, as other students in our study do; they talk only about students ‘that don’t do their work’, a point to which I shall return later in the paper.

The following boy was achieving at lower levels than other students and it is interesting to hear him talk about the ways he was supported by the practices of explanation and justification:

Juan: Most of them, they just like know what to do and everything. First you’re like “why you put this?” and then like if I do my work and compare it to theirs. Theirs is like super different ‘cos they know, like what to do. I will be like – let me copy, I will be like “why you did this?


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And then I’d be like: “I don’t get it why you got that.” And then like, sometimes the answer’s just like, they be like “yeah, he’s right and you’re wrong” But like – why?” (Year 2)

Juan also differentiates between high and low achievers without referring to such adjectives as ‘smart’ or ‘fast’, instead saying that some students ‘know what to do’. He also makes it very clear that he is helped by the practice of justification and that he feels comfortable pushing other students to go beyond answers and explain ‘why’ their answers are given. At Railside the teachers have carefully prioritized the message that each student has two important responsibilities – both to help someone who asks for help, but also to ask if they need help. Both are important in the pursuit of equity, and justification has emerged as a helpful practice in the learning of a wide range of students, particularly because the act of justification makes mathematical methods more transparent and learnable for different students.

Assigning Competence

An interesting and subtle approach that is recommended within the complex instruction literature is that of ‘assigning competence’. This is a practice that involves teachers raising the status of students that may be of a lower status in a group, by, for example, praising something they have said or done that has intellectual value, and bringing it to the group’s attention; asking a student to present an idea; or publicly praising a student’s work in a whole class setting. This practice was one that I could not fully imagine until I saw it enacted. My first awareness of it came about when a quiet Eastern European boy muttered something in a group that was dominated by two happy and excited Latina girls. The teacher who was visiting the table immediately picked up on it saying ‘Good Ivan, that is important’. Later when the girls offered a response to one of the teacher’s questions he said, ‘Oh that is like Ivan’s idea, you’re building on that’. He raised the status of Ivan’s contribution, which would almost certainly have been lost without such an intervention. Ivan visibly straightened up and leaned forward as the teacher reminded the girls of his idea. Cohen (1994) recommends that if student feedback is to address status issues, it must be public, intellectual, specific and relevant to the group task (p. 132). The public dimension is important as other students learn about the broad dimensions that are valued; the intellectual dimension ensures that the feedback is an aspect of mathematical work, and the specific dimension means that students know exactly what the teacher is praising.

The practice of ‘assigning competence’ is subtle and requires highly sensitive teachers. It is a practice that many good teachers employ, to a greater or lesser extent, without necessarily being aware of it or having a name for it. This practice is also an important part of the teachers’ commitment to equity and to the principle of showing what different students can do in a multifaceted mathematical context.


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Teaching Students to be Responsible for their Own Learning

A major part of the equitable results attained at Railside is the serious way in which teachers expect students to be responsible for each other’s learning. Many schools employ group work which, by its nature, brings with it an element of responsibility, but Railside teachers go well beyond this to ensure that students take the responsibility very seriously.

One way in which teachers nurture a feeling of responsibility is through the assessment system. Teachers grade the work of a group by, for example, rating the quality of the conversations groups have. The teachers also occasionally give group tests, which take several formats. In one version students work through a test together, but the teachers grade only one of the individual papers and that grade stands as the grade for all the students in the group. A third way in which responsibility is encouraged is through a practice of asking one student in a group to answer a follow up question after a group has worked on something. If the student cannot answer the question the teacher will leave the group to have more discussions and return to ask the same student again. In the intervening time it is the group’s responsibility to help the student learn the mathematics they need to answer the question. The teacher move of asking one member of a group to give an answer and an explanation, without help from their groupmates, is a subtle practice that has major implications for the classroom environment. This practice means that students are responsible to everyone in their group. In the following interview extract the students talk about this particular practice and the implications it holds:

Int: Is learning math an individual or a social thing?

Gisella: It’s like both, because if you get it, then you have to explain it to everyone else. And then sometimes you just might have a group problem and we all have to get it. So I guess both.

Bianca: I think both because individually you have to know the stuff yourself so that you can help others in your group work and stuff like that. You have to know it so you can explain it to them. Because you never know which one of the four people she’s going to pick. And it depends on that one person that she picks to get the right answer. (Year 2)

The students in the extract above make the explicit link between teachers asking any group member to answer a question, and being responsible for their group members. They also communicate an interesting social orientation that becomes instantiated through the mathematics approach, saying that the purpose in knowing individually is not to be better than others but so “you can help others in your group”. The four practices I have described so far – those of taking group roles, multidimensionality, assigning competence and encouraging responsibility are all part of the complex instruction approach. I will now review three other practices in which the teachers engage that are


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also critical to the promotion of equity. These relate to the challenge and expectations provided by the teachers.

Challenge and Expectations

High Cognitive Demand

The teachers at Railside provide a huge amount of support to students, making themselves available to students after school and in the evenings. In interviews the students talk at length about the support of the teachers. But the teachers also expect a lot from the students and give them complex, mathematically challenging work. Importantly the support that teachers give to students does not serve to reduce the cognitive demand of the work (Stein, Smith, Henningsen & Silver, 2000). The cognitive demand that is expected of all students is higher than other schools partly because the classes are heterogeneous and no students are precluded from meeting high level content, but teachers also enact a high level of challenge in their interactions with groups and their questioning.

A very important feature of the curriculum that teachers use, that would not be seen in the curriculum materials, is the act of asking follow up questions. The curriculum example analyzed in the longer version of this paper (Boaler, 2004) involves students finding a perimeter of a complex shape made up of algebra tiles. The perimeter is 10x +10 and the teacher asks the follow up question: ‘where is the 10?’. This is complex because the 10 does not exist in a particular place in the diagram and its detection involves an understanding of algebraic variables. The students are aware that the teachers keep levels of mathematical work high and they appreciate it. When we interviewed students and asked them what it takes to be a good teacher, many of them mentioned the high demand placed upon them, for example:

Ana: She has a different way of doing things. I don’t know, like she won’t even really tell you how to do it. She’ll be like, ‘think of it this way’. There’s a lot of times when she’s just like – ‘well think about it’ – and then she’ll walk off and that kills me. That really kills me. But it’s cool. I mean it’s like, it’s alright, you know. I’ll solve it myself. I’ll get some help from somebody else. It’s cool. (Year 3)

The following students, in talking about the support teachers provide, also refer to their push for understanding:

Int: What makes a good teacher?

John: Patience. Because sometimes teachers they just zoom right through things. And other times they take the time to actually make sure you understand it, and make sure that you actually pay attention. Because there’s some teachers out there who say: ‘you understand this?’ and you’ll be like “yes”. But you really don’t mean “yes” you mean “no”.


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And they’ll be like “OK” And they move on. And there’s some teachers that be like – they know that you don’t understand it. And they know that you’re just saying “yes” so that you can move on. And so they actually take the time out to go over it again and make sure that you actually got it, that you actually understand this time. (Year 2)

The students’ appreciation of the teachers’ demand was also demonstrated in our questionnaires. One of the questions started with the stem: ‘When I get stuck on a math problem, it is most helpful when my teacher …’ This was followed by answers such as ‘tells me the answer’ ‘leads me through the problem step by step’ and ‘helps me without giving away the answer’. Students could respond to each on a fourpoint scale (SA, A, D, SD). Almost half of the Railside students (47%) strongly agreed with the response: “Helps me without giving away the answer,” compared with 27% of students in the ‘traditional’ classes at the other two schools (n= 450, t = 4.257 ; df = 221.418; p


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When we asked in questionnaires: ‘How long (in minutes) will you typically work on one math problem before giving up and deciding you can't do it?’ the Railside students gave responses that averaged 19.4 minutes, compared to the 9.9 minutes averaged by students in traditional classes (n=438, t = 5.641; df = 142.110; p


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equal over the course of an entire life” (Anderson, p. 319)’ (from Post, 2004, p. 11). This conception shares some aspects of the version of equity I would like to propose. In observing Railside classes over many years I have come to appreciate what I will term as relational equity (Boaler, 2008). This conception of equity also goes further than students’ test results in schools and considers the ways students are acting in school and the ways they learn to regard and relate to one another. Unlike Anderson’s conception of equity, relational equity is about students’ relationships with other students and with subjects and with the intersection of the two, as I shall outline below. In proposing this conception of equity I am asking the question – what would it mean to teach students to act in equitable ways in classrooms? This question was raised, for me, through observations of the Railside classrooms and the behaviour of the teachers and students. Indeed it would be hard to spend years in the classrooms at Railside without noticing that the students were learning to treat each other in more respectful ways than is typically seen in schools and that ethnic cliques were less evident in the mathematics classrooms than they are in most schools. Further such behaviour did not just happen to take place in a mathematics classroom, it was fundamentally related to the students’ conceptions of and work within mathematics. I propose the term relational equity because it is about students’ relationships with each other developed through mathematical work as well as students’ relationships with the subject of mathematics. Such relationships led to more equitable achievement results at the school as well, I contend, to more equitable ways of working that students would take into their lives.

Richard Schweder, a cultural anthropologist and cultural psychologist talks about the importance of considering different perspectives on issues in the working of a democratic society:

It is often advantageous to have more than one discourse for interpreting a situation or solving a problem. Not only alternate solutions but multidimensional ones addressing “several orders of reality” or “orders of experience” may be more practical for solving complex human problems. (Schweder, 2003, p. 100)

The Railside students learn through their mathematical work that alternate and multidimensional solutions are important which leads them to value the contributions of the people offering such ideas. This is particularly important at Railside as the classrooms are multicultural and multilingual. It is commonly believed that students will learn respect for different people and cultures if they have discussions about such issues or read diverse forms of literature in English or Social studies classes. I propose that all subjects have something to contribute in the promotion of equity and that mathematics, often regarded as the most abstract subject removed from responsibilities of cultural or social awareness, has an important contribution to make. For the equitable relationships that Railside students developed and that I have discussed more fully in Boaler (2004) are


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only made possible by a conception of mathematics that values the contribution of different insights, methods and perspectives in the collective solving of a particular problem with a particular solution. The relational version of mathematics that these students learn in which they come to value different contributions to a problem and different relationships between mathematical methods enables the respectful working relationships I have set out. It seems to me that it is important to consider whether students are learning to respect students from different ethnic and cultural groups and genders in our schools today yet such concerns are not captured in notions of equity that are measured by test scores. Relational equity builds from the relations within subjects. As students learn to appreciate and make connections between different mathematical methods and ideas and to value the contributions of different perspectives on problems, given by students who think in quite different ways, they will also learn something extremely valuable about people, relations and ideas.

CONCLUSION

Railside School is not a perfect place the teachers would like to achieve more in terms of student achievement and the elimination of inequities, and they rarely feel satisfied with the achievements they have made to date, despite the vast amounts of time they spend planning and working. But research on urban schools and the experiences of mathematics students in particular tells us that the achievements at Railside are extremely unusual. In this paper I have attempted to convey the work of the teachers in bringing about the reduction in inequalities as well as general high achievement that they achieve. In doing so I hope also to have given a sense of the complexity of the relational and equitable system that they have in place (see also Boaler, 2009). People who have heard about the achievements of Railside have asked for their curriculum so that they may use it, but whilst the curriculum plays a part in what is achieved at the school it is only one part of a complex, interconnected system. At the heart of this system is the work of the teachers, and the many different equitable practices in which they engage.

REFERENCES

Anderson, E. S. (1999). What is the Point of Equality? Ethics, 109 (January 1999), 287 337.

Boaler, J. (1997). Experiencing School Mathematics: Teaching Styles, Sex and Setting. Buckingham: Open University Press.

Boaler, J. (2002a). The Development of Disciplinary Relationships: Knowledge, Practice and Identity in Mathematics Classrooms. For the Learning of Mathematics, 22(1), 42 47.


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Boaler, J. (2002b). Experiencing School Mathematics: Traditional and Reform Approaches to Teaching and Their Impact on Student Learning. (Revised and Expanded Edition). Mahwah, NJ: Lawrence Erlbaum Association.

Boaler, J. (2003). When Learning No Longer Matters: Standardized Testing and the Creation of Inequality. Phi Delta Kappan, 84(7), 502506.

Boaler, J. (2008). Promoting 'relational equity' and high mathematics achievement through an innovative mixed ability approach. British Educational Research Journal. 34 (2), 167194

Boaler, J. (2009). The Elephant in the Classroom: Helping Children Learn and Love Maths. London: Souvenir Press.

Boaler, J., & Brodie, K. (2004). The Importance of Depth and Breadth in the Analysis of Teaching: A Framework for Analysing Teacher Questions. Paper presented at the Psychology of Mathematics Education NA. Toronto, Ontario.

Cohen, E. (1994). Designing Groupwork. New York: Teachers College Press.

Cohen, E., & Lotan, R. (Eds.). (1997). Working for Equity in Heterogeneous Classrooms: Sociological Theory in Practice. New York: Teachers College Press.

Fendel, D., Resek, D., Alper, L., & Fraser, S. (2003). Interactive Mathematics Program: Integrated High School Mathematics. Berkeley, CA: Key Curriculum

Press.

Fiori, N. & Boaler, J. (2004). What Discussions Teach us about Mathematical Understanding: Exploring and Assessing Students’ Mathematical Work in Classrooms. Paper presented at the Psychology of Mathematics Education NA. Toronto, Ontario.

Horn, I.S. (2002). Learning on the job: Mathematics teachers’ professional development in the contexts of high school reform. Unpublished Ph.D. Dissertation, University of California, Berkeley.

Horn, I.S. (2005). Learning on the job: A situated account of teacher learning in two high school mathematics departments. Cognition & Instruction, 23(2).

Lee, C. D. (2001). Is October Brown Chinese? A Cultural Modeling Activity System for Underachieving Students. American Educational Research Journal, 38(1), 97141.

Maher, C., & Martino, A. M. (1996). The Development of the Idea of Mathematical Proof: A 5Year Case Study. Journal for Research in Mathematic Education, 27(2), 194214.

Picciotto, H. & Wah, A. (1994). Algebra: Themes, concepts, and tools. Aslip, IL: Creative Publications.


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Post, L. (2004). Working Toward Equity: Teacher Communities Changing Practices. Unpublished PhD Dissertation, Stanford University: Stanford, CA.

RAND, M. S. P. (2002, October). Mathematical Proficiency for All Students: Toward a Strategic Research and Development Program in Mathematics Education (Dru2773 Oeri). Arlington, VA: RAND Education & Science and Technology Policy Institute.

Schweder, R.A. (2003). Why Do Men Barbeque? Recipes for cultural psychology. Cambridge, MA: Harvard University Press, 74133.

Stein, M. K., Smith, M., Henningsen, M., & Silver, E. (2000). Implementing Standards Based Mathematics Instruction: A Case Book for Professional Development. New York: Teachers College Press.

Yackel, E., & Cobb, P. (1996). Sociomathematical Norms, Argumentation, and Autonomy in Mathematics. Journal for Research in Mathematics Education, 27(4), 458477.


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TEACHERS ASSESSING YOUNG CHILDREN'S MATHEMATICAL DEVELOPMENT: HOW CONFIDENT ARE THEY?

Janette Bobis

University of Sydney, Australia

Research has indicated that teachers’ knowledge of children’s mathematical thinking can be particularly influential in determining their instructional strategies and can potentially increase their abilities to cater for various levels of children’s mathematical understanding (Swafford, Jones and Thornton, 2000). This study sought to explore just how confident teachers are in their abilities to assess students’ mathematical development and the extent to which they perceived such knowledge impacted on their instructional decisionmaking. Twentytwo primary school teachers, who had participated in a professional development program that specifically focused on understanding and applying a theoretical framework of children’s cognitive development in number, were asked to rate their confidence in identifying students’ levels of development according to the learning framework. Followup interviews elicited explicit information about each teacher’s ability to utilise the framework to assess individual student’s mathematical development and to plan appropriate instruction as a consequence. Teachers’ selfratings of their abilities to utilise a theoretical framework to assess student progress and undertake subsequent planning for instruction were generally lower than what they were able to demonstrate in reality. Interviews revealed that the majority of teachers held extensive knowledge about students’ mathematical development and could clearly articulate a trajectory of learning appropriate for each child’s level of development.

INTRODUCTION

It has only been since Shulman delivered his seminal paper on teacher knowledge in 1986, in which he delineated three categories of knowledge—subject matter content knowledge, pedagogical content knowledge, and curricular knowledge, that the study of teacher knowledge has really become a central focus of educational researchers and policy makers. Since then, many educators and researchers have advanced their own frameworks and interpretations of various components of teacher knowledge (e.g., Hill, Rowan and Ball, 2005). Importantly, attention has changed from looking solely at what content knowledge teachers possess to the quality of their understanding of how children learn that content (including major growth points in their development of understanding) and of the specific pedagogy needed to teach it (Hill, Ball and Schilling, 2008).

When studying teachers’ knowledge of mathematics it is essential to also consider teachers’ perceptions of that knowledge (Li and Kulm, 2008). While distinct from knowledge (Thompson, 1992), teachers’ perceptions of, and confidence in, their knowledge seems to particularly impact on their classroom instruction (Cai, Perry and


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Wong, 2007). Without confidence in their knowledge to determine students’ mathematical development, teachers’ may question their abilities to plan appropriate instruction.

‘Confidence’ is a dimension of attitude that has been studied quite extensively in relation to teachers’ mathematical content knowledge, particularly in primary and middle school teachers (Beswick, Watson and Brown, 2005), but there is comparatively little research that explores teachers’ confidence to assess students’ mathematical development and to plan appropriate instruction as a consequence. The explicit assessment of teachers’ confidence to apply their knowledge of how children learn mathematics was an important component of a larger study designed to explore the impact of a professional development program on teachers’ knowledge and classroom practices. Before presenting the specific research questions of the study relating to teacher confidence, background to the professional development program is required to fully appreciate the context for the study and research questions.

BACKGROUND TO THE STUDY

The Count Me In Too numeracy program is an ongoing professional development initiative of the Department of Education and Training in New South Wales (NSWDET, 2007), Australia. Its aims are to help teachers understand children’s mathematical development and to improve children’s achievement in mathematics. The Count Me In Too program began in 1996 as a pilot program involving 13 schools and gradually expanded to involve nearly 1700 primary schools over a tenyear period across the state. Key aspects of the program include the Learning Framework In Number (LFIN) and a diagnostic interview or Schedule in Early Number Assessment (Wright, Martland and Stafford, 2006). The LFIN is used by teachers to not only identify the level of development each child has attained but provides instructional guidance as to what each student needs to work towards. A stimulus for the current study, was the need to know what teachers understand about the LFIN and how it impacts on their teaching and assessment practices. Importantly, the LFIN has had a major impact on the development of syllabus and curricula documents throughout Australia, New Zealand and a growing number of other regions in the world. Hence, the implications of this study have potentially quite substantial application.

The Learning Framework In Number

Learning frameworks, also known as progress maps or learning trajectories provide a description of skills, understandings and knowledge in a sequence in which they typically occur, thus giving a virtual picture of what it means to progress through an area of learning. Thus they provide a pathway or map for monitoring individual development over time. A student’s location on a framework can be utilised as a guide to determining the types of learning experiences that will be most useful in meeting the student’s individual needs at that particular stage in their learning. A number of professional


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development programs now exist that utilize such theoretical frameworks with the aim of increasing teachers’ understanding of children’s mathematical thinking (e.g., Bobis, Clarke, Clarke, Thomas, Wright, YoungLoveridge and Gould, 2005; Van den Heuvel Panhuizen, 2001).

The Count Me In Too Learning Framework In Number was initially developed by Wright (1994) and has since undergone further development through the impact of a wide range of research in early number (e.g., Gravemeijer, 1994; Mulligan and Mitchelmore, 1997). In brief, the LFIN consists of five key and interrelated components:

1. Building addition and subtraction through counting by ones

2. Building addition and subtraction through grouping

3. Building place value through grouping

4. Building multiplication and division through equal counting and grouping

5. Building fractions through sharing and partitioning

The LFIN provides a description of the knowledge and skills characterising major stages of development in each of these components. Teachers use these stage descriptions to profile their students’ knowledge in each key component. Such information then provides instructional guidance as to what each student needs to progress. An important step in a teacher’s ability to utilise the framework in their instructional decisionmaking is their understanding of how all components are interrelated. A visual representation of the Framework is presented in Figure 1 and a more detailed description of the LFIN is available in Wright, Martland and Stafford (2006) and at the NSW Department of Education and Training website ( www.curriculumsupport.education.nsw.gov.au/countmein/).

http://www.curriculumsupport.education.nsw.gov.au/countmein/


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Figure 1. Learning Framework In Number model showing five major knowledge components and important prerequisite number skills.

THE STUDY

The overall aim of the larger study was to explore teacher knowledge of the Learning Framework in Number from the Count Me In Too numeracy program and the impact this knowledge has on their instructional decision making for teaching mathematics. This paper focuses on a component of the larger study, namely, teachers’ perceptions/confidence of their knowledge of the Framework and the impact these perceptions have on their ability to assess learning and plan for instruction. In particular, the following research questions were addressed:

1. What are teachers’ perceptions about the extent of their knowledge of the Learning Framework In Number [LFIN]?

2. How confident do teachers feel about identifying children’s levels of mathematical development on the LFIN?

3. Can teachers use their knowledge about children’s mathematical development as indicated on the LFIN to plan appropriate instruction?

Information for the larger study was gathered from three main sources—survey, interviews and teaching documents. During the survey component, teachers were asked to rate themselves on a number of aspects. Three aspects related to their understanding of the LFIN, their confidence using it to assess children’s mathematical development and the extent to which they perceived this knowledge impacted on their ability to plan appropriate instruction. Following the survey, teachers whose responses were considered representative of the sample, were invited to be interviewed. Each of these teachers had


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been initially invited to participate in the study due to their ongoing involvement in the Count Me In Too numeracy professional development program that was operating in their schools. Only data from the survey and interview component (interviews and teaching documents) related specifically to the issue of teacher confidence for these teachers are reported here.

Participants and Procedure

Twentytwo teachers (21 female and 1 male) drawn from three different schools agreed to participate in the interview component—8 teachers from School A and 7 teachers from both School B and School C. Each interview took approximately 45 – 60 minutes and was conducted in a private office within the school grounds during school hours. Relief teaching was provided for the duration of the interview so that teachers were not inconvenienced by time taken for the interview.

The interviews established background biographical details for each interviewee before seeking information specifically related to professional learning and the implementation of the numeracy program and the LFIN. Teachers were then asked about their confidence concerning the identification of individual students’ abilities using the LFIN and the subsequent planning for student instruction. The interviews were digitally recorded and later transcribed for the purpose of analysis. It was requested that teachers bring documents such as school management plans and individual or collaborative teacher programs to the interview to help support their oral explanations of planning and teaching practices.

Analysis

Background data on each survey respondent were collated and descriptive statistics were used to analyse the items requiring teachers to provide personal ratings for certain aspects of their knowledge and utilisation of the LFIN. Interview data were transcribed and read for emerging themes that specifically related to teacher confidence surrounding the LFIN and instruction.

Documents in the form of school management plans and individual teacher programs were analysed in conjunction with interview data since all documents provided by teachers were intended to support, elaborate and verify their responses to interview questions. Teacher programs and lesson plans, when available, were analysed to determine the type and level of impact the LFIN had on planning for student learning at the school and classroom levels.

RESULTS AND DISCUSSION

Background information for each interviewee, along with their selfratings on the survey


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Interviewee Grade Approx.

Months/Yr using CMIT

Selfrating for perceived

understanding of LFIN (1 to 4)

Selfrating for confidence to use LFIN to assess students’ (1 to 4)

Selfrating for extent LFIN perceived to

impact on instruction (1 to 4)

1. Jane 5 & 6 5 months 2 3 3

2. Kathy K 8 years 3 3 3

3. Roberta 3 & 4 5 months 2 2 1

4. Lilly 1 1 year 3 3 3

5. Lana 3 & 4 1 year 2 3 2

6. Ann K 2 years 3 3 4

7. Kate 3 & 4 1 year 1 1 2

8. Mark 3 & 4 1 year 2 3 3

9. Alyson 3 6 years 3 3 2

10. Naomi 1 5 months 2 1 2

11. Tania 1 & 2 8 years 3 3 3

12. Mandy 2 4 years 3 3 3

13. Maxine 4 & 5 6 years 1 3 3

14. Katie 5 & 6 5 years 3 4 3

15. Narelle 1 1 year 1 3 2

16. Robyn 2 2 years 3 3 3

17. Vera K 2 years 3 3 3

18. Violet K 2 years 2 2 3

19. Kristen 1 & 2 2 years 1 1 1

20. Erica 1 5 months 2 3 2

21. Natalie K & 1 5 months 3 4 3

22. Christine 3 10 years 3 3 4

Table 1. Summary of background information and teachers’ selfratings for understanding and confidence using the LFIN to assess and plan for instruction from survey results

component for their understanding and confidence using the LFIN are presented in Table 1.


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As can be seen from the table, teachers’ experience implementing the numeracy program (and using the LFIN) ranged from just 5 months to 10 years. The mean number of years experience of the entire sample was slightly less than 3 years (2 years 9 months). While the program had been implemented in each school for at least 3 years, teachers’ experience differed for various reasons; some had been exposed to the program in a previous school, others had begun employment after its initial introduction at their current school and some schools had chosen to ‘rollout’ the implementation starting with more junior classes in the first year and gradually extending the program to the higher grades in the second and third years of the program. In research terms, it would have been more ideal if there was greater consistency in teachers’ experience with the program, but that was not an initial aim of the study and nor is it realistic in a school context.

Table 2 summaries the number and percentage of interviewees selecting each level rating on the three aspects of teacher confidence. It can be seen from the table, that no interviewees rated their understanding of the LFIN at the highest level (Level 4) and only 2 teachers considered their ability to assess students’ mathematical ability or to plan appropriate instruction using the LFIN to be Level 4. The majority of respondents rated themselves at Level 3 for each aspect dealing with teacher confidence on the survey.

Rating Understanding of LFIN

Confidence using LFIN to assess students

Extent LFIN impacts on instruction

1 3 (14%) 2 (9%) 2 (9%)

2 7 (32%) 2 (9%) 6 (27%)

3 11 (50%) 15 (68%) 12 (55%)

4 0 (0%) 2 (9%) 2 (9%)

Table 2. Number (and percentage) of interviewees selecting each level rating on the three aspects of confidence

An initial purpose of the interviews was to validate and elaborate on information obtained from the survey data collected. In particular, interviews were a critical means by which teachers’ reasons for their personal ratings about the LFIN and its perceived impact on their knowledge and instructional decisionmaking could be verified. Individual and gradeteam teaching programs were presented by interviewees to help explain and justify descriptions of their planning and teaching practices. Reference to these documents is integrated into the discussion of interview data.

Overall, interviewees expressed reluctance to give themselves an ‘Excellent’ or Level 4 rating for any of the aspects they had been asked to rate. Two Year 3/4 teachers (Roberta and Kate) commented that with less than a years experience working with the Framework, they “still had a lot to learn”. Mark thought that his understanding was currently a Level 2, but expected “given time, I will be able to delve into it more”.


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The majority of teachers rated their confidence using the LFIN to identify children’s stages of development as a Level 3. Generally, teachers agreed with Jane that “I haven’t done it enough to be sure” and Mark that “while I’m at least a Level 3 for confidence, I’m heading for a Level 4—I just need another year to consolidate what I’ve learnt”. Kate also thought that another year implementing the program would ensure an increase in her confidence. Narelle, a Year 1 teacher who had been implementing the numeracy program for approximately one year, explained her confidence in using the LFIN had grown enormously since she had used it to assess a second group of students. Hence, she rated her confidence using the Framework to assess her students as a Level 3. She only rated her understanding of the LFIN a Level 1 because she felt “there is so much more to learn about it—and I’m learning new things all the time”. As was the case for many interviewees, the perception that there is still much more to learn about the Framework and how to use it to guide instruction seemed to weigh heavily on teachers’ minds— resulting in conservative selfassessments of their knowledge about the LFIN and their interpretation of its use.

More than half the interviewees indicated that “more time” working with the LFIN as part of the numeracy program would allow them to become more “confident with my understanding of the Framework and how to implement it in the classroom”. The initial survey data provides some support for this belief; there was a clear trend linking the length of time a teacher had been utilising the Count Me In Too program to their self rated confidence level. That is, the more time teachers indicated that they had been implementing the program, the more confident they felt about their understanding of the LFIN and their ability to use it to guide their assessment and instructional decision making. However, the fact that so many teachers with one year or less experience implementing it also rated themselves at Level 3, indicates that ‘time’ alone is not the only factor impacting on teachers’ confidence levels. ‘Time’ by itself is not considered the determining factor for a teacher’s improved confidence level. Rather, the professional support at the school and, in particular, the stage/grade level, was considered more influential. This was evident when interviewees explained their selfratings for the extent to which the LFIN was perceived to impact upon their instruction.

The degree to which the LFIN was perceived to impact on classroom instruction was most evident when teachers discussed their program and lesson planning strategies. The majority of K4 teachers considered “[LFIN] activities are embedded into the number programs—they are not seen as something separate”. Evidence of this integration was found in the teachers’ programs. Kindergarten teachers, Violet and Vera presented their written programs to exemplify the explicit links contained in it to the LFIN. Violet explained that a programming “proforma” was used throughout the infant classes to provide some consistency within stages and that the Framework was used to “gear activities to specific groups of students”. In addition to syllabus outcomes, specified content and processes, explicit reference to aspects of the Framework were listed for the


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focus of each lesson and accompanied by lists of activities intended for varying student ability levels as indicated on the LFIN.

Not all programs for each teacher explicitly referred to strategy development named on the LFIN, but the learning experiences and activities followed a sequential development akin to the knowledge, skill and strategy development outlined on the LFIN. Additionally, named activities and resources in the programs made direct reference to curriculum resources that have the LFIN embedded in their structure (NSW DET, 1999; 2003). Mandy commented:

The Framework is the